## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 3, Number 2 (2001), 341-354.

### Cores of spaces, spectra, and {$E\sb \infty$} ring spectra

#### Abstract

In a paper that has attracted little notice, Priddy showed that the Brown-Peterson spectrum at a prime $p$ can be constructed from the $p$-local sphere spectrum $S$ by successively killing its odd dimensional homotopy groups. This seems to be an isolated curiosity, but it is not. For any space or spectrum $Y$ that is $p$-local and $(n_0-1)$-connected and has $\pi_{n_0}(Y)$ cyclic, there is a $p$-local, $(n_0-1)$-connected "nuclear" CW complex or CW spectrum $X$ and a map $f: X\to Y$ that induces an isomorphism on $\pi_{n_0}$ and a monomorphism on all homotopy groups. Nuclear complexes are atomic: a self-map that induces an isomorphism on $\pi_{n_0}$ must be an equivalence. The construction of $X$ from $Y$ is neither functorial nor even unique up to equivalence, but it is there. Applied to the localization of $MU$ at $p$, the construction yields $BP$.

#### Article information

**Source**

Homology Homotopy Appl., Volume 3, Number 2 (2001), 341-354.

**Dates**

First available in Project Euclid: 13 February 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1139840257

**Mathematical Reviews number (MathSciNet)**

MR1856030

**Zentralblatt MATH identifier**

0987.55009

**Subjects**

Primary: 55P15: Classification of homotopy type

Secondary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)

#### Citation

Hu, P.; Kriz, I.; May, J. P. Cores of spaces, spectra, and {$E\sb \infty$} ring spectra. Homology Homotopy Appl. 3 (2001), no. 2, 341--354. https://projecteuclid.org/euclid.hha/1139840257